finding base of quadratic system

The roots of the quadratic equation x^2-12x+37=0 are 5 and 8. Find the base system in which this equation is written

If the roots are 5 and 8, then the equation can be written as
(((x^2 -(5+8)x + (5*8) = 0}}}
so in what base is 5+8 = 12 and 5*8 = 37?
The base must be greater that 8 since one of the solutions is 8.
The base is greater than 10 since 5*8 base 10 is 40 and our product is 37
Try 11. Hey, it works!
5+8 base 11 = 12 and (5*8) base 11 = 37.

How they say the base should be greater than 8 and also than 10..thank you in advance

Since one of the roots is 8, the base must be greater than 8. What if the base were less than 8? Then there is no digit 8! A value of decimal 8 in base 8 would be 10!

Similar logic can be applied for saying that the base must be greater than 10. 5*8=40 in base 10. But the result is 37. If the base were 10 or less, the result is 40 or more. What is 40 base 10 in base 9? (44) See the direction the digits are going?